On the eigenvalues of quaternion matrices |
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| Authors: | F O Farid Qing-Wen Wang Fuzhen Zhang |
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| Institution: | 1. Kelowna, B.C. , V1W 4T4 Canada farid-f@shaw.ca;3. Department of Mathematics , Shanghai University , Shanghai 200444, China;4. Division of Math, Science and Technology , Nova Southeastern University , 3301 College Ave, Fort Lauderdale, Florida 33314, USA;5. School of Mathematics and Systems Science , Shenyang Normal University , Shenyang, Liaoning Province 110034, China |
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| Abstract: | This article is a continuation of the article F. Zhang, Ger?gorin type theorems for quaternionic matrices, Linear Algebra Appl. 424 (2007), pp. 139–153] on the study of the eigenvalues of quaternion matrices. Profound differences in the eigenvalue problems for complex and quaternion matrices are discussed. We show that Brauer's theorem for the inclusion of the eigenvalues of complex matrices cannot be extended to the right eigenvalues of quaternion matrices. We also provide necessary and sufficient conditions for a complex square matrix to have infinitely many left eigenvalues, and analyse the roots of the characteristic polynomials for 2?×?2 matrices. We establish a characterisation for the set of left eigenvalues to intersect or be part of the boundary of the quaternion balls of Ger?gorin. |
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| Keywords: | Brauer's theorem Ger?gorin theorem left eigenvalue quaternion quaternion matrix right eigenvalue |
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